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Optimization of Grafting an Artery

A surgeon is faced with the problem of grafting an artery. She wishes to minimize the resistance to the resulting flow.
The resistance R to the laminar flow in a pipe L is given by Poiseuille's law: R = L/ r^4

This is where L and r are the length and radius respectively of the pipe. The graft must run from a main artery of radius r_1=0.5 cm to a point 5 cm from the main artery, using a connecting artery of radius r_2=0.46.
Coordinate the problem by assuming that the main artery runs along the x-axis, so the connecting artery needs to run to the point (10,5) and the graft occurs at (x,0).

Answer the following questions in terms of x.

What is the length of an artery running from (0,0) to (x,0)?
What is the resistance of the artery running from (0,0) to (x,0)?
What is the length of an artery running from (x,0) to (10,5)?
What is the resistance of the artery running from (x,0) to (10,5)?
Write the total resistance from the origin to the point (10,5) as a function of x, assuming that the graft occurs at (x,0) and assuming that the resistance at the graft itself is negligible.
What value of x minimizes this total resistance?
x_{opt} = ?

Solution Preview

What is the length of an artery running from (0,0) to (x,0)?
The length is the distance from the point (0,0) to the point (x,0) (assuming that the artery is straight). The distance from the point (0,0) to the point (x,0) is |x|, the absolute value of the number x.

What is the resistance of the artery running from (0,0) to (x,0)?
Since the artery, running from (0,0) to (x,0) is the main artery of radius r_1=0.5 cm, the resistance R_1 is R_1= L/ r_1^4=|x|/(0.5)^4=16 |x|.

What is the length of an artery running from (x,0) to (10,5)?
Again, assuming that the ...

Solution Summary

The solution finds optimal values for the problem of minimization of resistance to the laminar flow through an artery. For this we introduce coordinates and use Poiseuille's law for the resistance to the laminar flow in a pipe.

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